Last time we looked at the imprecise pairing of chaoplexity with scientific modeling and dubious use of quantum mechanics to explain a spontaneously appearing universe. My final example of suspect mathematics in science is even more metaphysical, string theory.

In the first half of the 20th century it must have appeared that the fundamental nature of matter was finally elucidated with Niels Bohr’s model of the atom with three readily understandable subatomic particles: electrons, protons, and neutrons. Unfortunately, that hope was brief since astronomers were already finding other particles in cosmic radiation. Later research using particle accelerators led to the unsightly Standard Model of Particle physics wherein subatomic particles like the proton are made up of still smaller particles called quarks of which there are dozens with odd names like the down quark or the charmed antiquark; coming in ‘flavors’ arbitrarily called red, green, and blue; and with spin of 0, 1, or 2.3

Such a disconcertingly complex outcome led many physicists to seek a still ‘smaller’ and simpler explanation underlying this particle ‘zoo.’ At that point string theory comes on the scene. The general theory is simple enough – particles are not points, but “string-like” and can be (1) stretched like rubber bands; more energy when stretched and less when contracted, and (2) vibrate like rubber bands. With work physicists and mathematicians were able to make string theory work by joining it to supersymmetry leading to the more coherent superstring theory. Here at last was a theory that could unify physics, a theory of everything; explaining  all of the particles, the forces, and the laws of motion, and accommodating special relativity and quantum theory.4

However there are problems. First superstring theory is actually many equally coherent theories, each requiring more than the commonly accepted  four dimensions – in fact 10 in all (the remaining six being tiny curled up dimensions). String theory depends on only one constant, but requires many additional seemingly arbitrary constants to explain the standard model. The theory is contingent on supersymmetry which is not visible in the natural world. The theory itself appears to be untestable. Last is the question of how the differences between unified particles and forces is to be explained.5

Some physicists are dubious of superstring theory. Richard Fenyman dislikes the tendency to explain or discount anything inconsistent with the theory. Sheldon Glashow scoffs that it cannot be demonstrated and has not led to a single experimental prediction.6 Lee Smolin bemoans the incredible resources diverted to this theory to the exclusion of other research based on the scientific community’s rigid belief in the theory or ‘groupthink’, referencing Kuhn’s book on scientific paradigms at one point. Others believe string theory’s greatest strength is its beauty, suggesting it qualifies as aesthetics.7

None of this is intended to diminish science which for the most part is the best system for identifying “truth” known to humanity. However my goal is to remind readers that the mathematics underpinning science and some resulting theories are not certain, some not even in an empirical context. Next time we look at one more concern with science as certainty, issues of connection.


1Horgan, John, The End of Science, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1996. ISBN 0-201-62679-9, page 191.

2Ibid. Page 202.

3Hawking, Stephen, A Brief History of Time, Bantam Books, New York, 2009. ISBN: 978-0-307-29117-2, pages 86-89.

4Smolin, Lee, The Trouble with Physics, Houghton Mifflin Company, Boston, 2007. ISBN: 978-0-618-91868-3, pages 103-112.

5Ibid., page 117-123.

6Ibid., page 125.

7Horgan, John, The End of Science, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1996. ISBN 0-201-62679-9, page 70.

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“God used beautiful mathematics in creating the world.”– Paul Dirac, Nobel Laureate in Physics, 1933.      “



In the last post I noted that mathematics developed independently of science over thousands of years until the Age of Reason when several great minds yoked them initiating a revolution in science. However, by the early 20th century, cracks began to show both in the proposition that mathematics is absolute and in the application of increasingly abstract mathematics to empirical reality. The doubts inherent in these mathematical speculations and modeling is the subject of this post.

Our first example is the area of chaos theory and complex system analysis which John Horgan labels chaoplexity.1 While basic science depends on the assumption of the uniformity of simple systems in generating scientific laws, most of reality is complex. So while we can predict the product of the mixing of two chemicals in a test tube it is unclear what this tells of about chemical reactions occurring in Earth’s primordial soup. On the face of it, chaos theory’s central tenant that highly complex systems inevitably lead to unpredictability appears to be logically inconsistent with predictive modeling.

In addition scientific models are prone to structural problems including speculative assumptions and bias in the choice of inputs. Nancy Cartwright, a philosopher of science, considers numerical models nothing more than “a work of fiction.”2 Nonetheless experts constantly forecast future events based on this method including portentous phenomena such as climate change and the course of pandemics.

A second example is quantum mechanics and uncertainty, which is inscrutable at best and irrational at worst. Still some physicists use the probabilistic nature of matter in space-time implied by this theory to argue that even if nothing at all existed before the big bang, there was an infinitesimal chance that a singularity would appear spontaneously. Since there was an infinite period in which this could occur, the appearance of the universe from naught is a reasonable explanation (everything just came from nothing!). This too seems illogical since there would have been no time and no environment in which a spontaneous event could occur.

This theory does not appear empirical at all, rather a mathematical labyrinth requiring assumptions and contortions which are of course beyond non-mathematicians. In fact, this theory seems closer to metaphysics than strict science,and lay persons are expected to accept its “truth” on faith in the knowledge of the experts, a circumstance hauntingly reminiscent of the assertion of priests in early religions.

(continued next post)

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“If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics.”– George Santayana.                `               `



Last time we looked at whether mathematics corresponds to absolute truth; today we investigate mathematics as a tool of science particularly as the substructure of scientific certainty. Nowadays we see math as so integral to science that we might think they developed in lockstep, but that is not the case. Mathematics likely started as counting possessions such as fingers, children, goats, or coins; followed by the geometry necessary for land measurement and construction. At that time, science was predominantly observational – e.g. identification of the constellations and planets and the four ancient elements of earth, water, air, and fire – or speculative as in the case of Democritus’ atomism.

Some ancient geniuses transposed common mathematics onto the mystery of nature most famously in an increasing understanding of the motion of the planets and sun (Thales predicted an eclipse in the sixth century B.C.E.) and on the harmonics of stringed instruments (Pythagoras; also the sixth century B.C.E.). But successors failed to follow up on their insights, so in fact the greatest scientist of the ancient world, Aristotle, studied zoology and botany only by observation and description; while its greatest mathematician, Archimedes, took mathematics much further, but mostly for technology rather than for the analysis of nature.

Successive cultures in Rome, Arabia, India, and even medieval Europe advanced in pure and applied mathematics, but failed to identify its utility in elucidating nature. That seems to appear suddenly in the works of Copernicus, Kepler, and particularly Galileo who rejected scholastic views of knowledge and subjected observational and experimental data to mathematical analysis in formulating theories. Arguably their insight that mathematics can explain data was their greatest contribution to science and one of the great feats of humanity. Newton, Pascal, Lavoisier, Faraday, Einstein and countless others followed, all using mathematics as the knife by which to dissect out the hidden structure of reality.

Thereafter  for centuries, mathematics and science grew in parallel without impediment until coming up against two fundamental challenges. First, mathematics itself showed defects as outlined in the last blog. Second, in the desire for solid foundations for scientific theories, mathematics was overstretched to fit some theories, and increasingly modified or invented merely to permit models of nature that transcend any experience of reality at all. It is this latter trend in the relationship of science to mathematics that is most disconcerting with respect to its certainty. Next time we will look at some important examples including (1) chaos theory and complex systems, (2) quantum mechanics, uncertainty, and the spontaneous appearance of matter, and (3) string theory.

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“As far as the laws of mathematics refer to reality they are not certain; and as far as they are certain, they do not refer to reality.”– Albert Einstein.

In our investigation of science as certainty, we arrive now at the complementary discipline of mathematics. If we accept the definition of science as “a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws,”1 then mathematics appears to be the first science methodically uncovered by the ancients. This seems confirmed by the Webster definition of mathematics as “the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities…”2 Alfred North Whitehead goes somewhat further in defining mathematics as “the science concerned with the logical deduction of consequences from the general premises of all reasoning,”3 though for our purposes we will use the use the stricter Webster definition.

In today’s blog we examine the certainty of mathematics while next time we will look at mathematics as a scientific tool. Grade school mathematics particularly addition and subtraction appear to be the pinnacle of unquestionable truth, but remain difficult to prove philosophically. The analytic philosophers in the early 19th century worked tirelessly at demonstrating that mathematics could be validated using only rigorous logic. However this  effort proved hopeless once Kurt Godel developed his ‘incompleteness theorem’ showing that any system that proves all true statements, also permits paradoxes that make no sense (such as “this sentence is false”), whereas any tinkering to the system to eliminate paradoxes results in some true statements no longer being demonstrable.

Meanwhile Gregor Cantor demonstrated that rules of infinity broke basic rules of mathematics, effectively proving that two unequal numbers can be equal. By mapping infinite series such as all integers, against all even numbers, he showed that while there are clearly more integers than even numbers, there are in fact an infinite number in each series. .

Other fields in mathematics also show areas of concern. Euclidean geometry looks ironclad, but  late 19th century mathematicians discovered non-Euclidean forms of geometry that were equally coherent, but gave different results (consider a triangle projected on a sphere versus a plane). Worse yet, some  scientists argue Euclidean geometry is not verifiable in the real world at all.

Chaos theory shows that unpredictable results or non-linearity occur in complex mathematical systems, thus undermining the presumption that all mathematic relations are absolute. In fact unpredictability is predictable in such systems, which appears inconsistent with our usual understanding of mathematics.

Also troublesome is meta-mathematics; which questions what numbers are (Platonic, self-existing, ideas vs. formalistic or logic-based entities) and the sublime question of whether mathematics is discovered or invented by the human mind (Einstein believed the latter). Such fundamental questions do not diminish the practical nature of mathematics, but do subvert our trust in its absolute certainty. As Morris Kline wrote in The Loss of Certainty in 1980, “It behooves us therefore to learn why, despite its uncertain foundations and despite the conflicting theories of mathematicians, mathematics has proved to be so incredibly effective.”4


1Webster’s New Universal Unabridged Dictionary, Barnes & Noble, Inc. 2003. ISBN 0-7607-4975-2, p. 1716 – definition 1.

2Webster’s New Universal Unabridged Dictionary, Barnes & Noble, Inc. 2003. ISBN 0-7607-4975-2, p. 1186 – definition 1.

3Fadiman, Clifton, Editor, The Treasury of the Encyclopaedia Britannica.Viking Penguin, New York, 1992. ISBN 0-670-83568-4, page 659.

4Ferris, Timothy, Editor, The World Treasury of Physics, Astronomy, and Mathematics. Little, Brown, and Co., Boston, 1991. Page 525.

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